Intuitionistic Fuzzy Topology and Intuitionistic Fuzzy Preorder
 Author: Yun Sang Min, Lee Seok Jong
 Publish: International Journal of Fuzzy Logic and Intelligent Systems Volume 15, Issue1, p79~86, 25 March 2015

ABSTRACT
This paper is devoted to finding relationship between intuitionistic fuzzy preorders and intuitionistic fuzzy topologies. For any intuitionistic fuzzy preordered space, an intuitionistic fuzzy topology will be constructed. Conversely, for any intuitionistic fuzzy topological space, we obtain an intuitionistic fuzzy preorder on the set. Moreover, we will show that the family of all intuitionistic fuzzy preorders on an underlying set has a very close link to the family of all intuitionistic fuzzy topologies on the set satisfying some extra condition.

KEYWORD
Intuitionistic fuzzy topology , Intuitionistic fuzzy preorder

1. Introduction
The theory of rough sets was introduced by Pawlak [1]. It is an extension of set theory for the research of intelligent systems characterized by insufficient and incomplete information. The relations between rough sets and topological spaces have been studied in some papers [2–4]. It is proved that the pair of upper and lower approximation operators is a pair of closure and interior of a topological space under a crisp reflexive and transitive relation.
The concept of fuzzy rough sets was proposed by replacing crisp binary relations with fuzzy relations in [5]. Furthermore in [6], the axiomatic approach for fuzzy rough sets were provided. In [7], the authors presented a general framework for the research of fuzzy rough sets in which both constructive and axiomatic approaches are used.
In [8], the authors showed that there is one to one correspondence between the family of fuzzy preorders on a nonempty set and the family of fuzzy topologies on this set satisfying certain extra conditions, and hence they are essentially equivalent.
This paper is devoted to finding relationship between intuitionistic fuzzy preorders and intuitionistic fuzzy topologies. For any intuitionistic fuzzy preordered space, an intuitionistic fuzzy topology will be constructed. Conversely, for any intuitionistic fuzzy topological space, we obtain an intuitionistic fuzzy preorder on the set. Moreover, we will show that the family of all intuitionistic fuzzy preorders on an underlying set has a very close link to the family of all intuitionistic fuzzy topologies on the set satisfying some extra condition.
2. Preliminaries
Let
X be a nonempty set. Anintuitionistic fuzzy setA is an ordered pairA = (µ_{A} ,ν_{A} ).where the functions
µ_{A} :X →I andν_{A} :X →I denote the degree of membership and the degree of nonmembership, respectively andµ_{A} +ν_{A} ≤ 1 (See [9]). Obviously every fuzzy setµ inX is an intuitionistic fuzzy set of the form .Throughout this paper,
I ⊗I denotes the family of all intuitionistic fuzzy numbers (a ,b ) such thata ,b ∈ [0, 1] anda +b ≤ 1, with the order relation defined by(
a ,b ) ≤ (c ,d ) iffa ≤c andb ≥d .And IF(
X ) denotes the family of all intuitionistic fuzzy sets inX , and ‘IF’ stands for ‘intuitionistic fuzzy.’Any IF set
A = (µ_{A} ,ν_{A} ) onX can be naturally written as a functionA :X →I ⊗I defined byA (x ) = (µ_{A} (x ),ν_{A} (x )) for anyx ∈X .For any IF set
A = (µ_{A} ,ν_{A} ) ofX , the valueπ_{A} (x ) = 1 −µ_{A} (x ) −ν_{A} (x )is called an
indeterminancy degree (orhesitancy degree ) ofx toA (See [9]). Szmidt and Kacprzyk [10] callπ_{A} (x ) anintuitionistic index ofx inA . Obviously0 ≤
π_{A} (x ) ≤ 1, ∀x ∈X .Note
π_{A} (x ) = 0 iffν_{A} (x ) = 1 −µ_{A} (x ). Hence any fuzzy setµ_{A} can be regarded as an IF set (µ_{A} ,ν_{A} ) withπ_{A} = 0.Definition 2.1 ( [11]). An IF setR onX ×X is called anIF relation on X . Moreover,R is called(i) reflexive if R(x, x) = (1, 0) for all x ∈ X, (ii) symmetric if R(x, y) = R(y, x) for all x, y ∈ X, (iii) transitive if R(x, y) ∧ R(y, z) ≤ R(x, z) for all x, y, z ∈ X.
A reflexive and transitive IF relation is called an
IF preorder . A symmetric IF preorder is called anIF equivalence . An IF preorder onX is called anIF partial order if for anyx ,y ∈X ,R (x ,y ) =R (y ,x ) = (1, 0) implies thatx =y . In this case, (X ,R ) is called anIF partially ordered space . An IF preorderR is called anIF equality ifR is both an IF equivalence and an IF partial order.Remark 2.2. R ^{−1} is called theinverse ofR ifR ^{−1} (x ,y ) =R (y ,x ) for anyx ,y ∈X . IfR is an IF preorder, so isR ^{−1} .R^{C} is called thecomplement ofR ifR^{C} (x ,y ) = (ν _{R(x,y)} ,µ _{R(x,y)}) whereR (x ,y ) = (µ _{R(x,y)} ,ν _{R(x,y)}). It is obvious thatR ^{−1} ≠R^{C} .Definition 2.3 ( [12]). LetR be an IF relation onX . Then the two functions , : IF(X ) → IF(X ), defined byare called the
upper approximation operator and thelower approximation operator onX , respectively. Moreover, (X ,R ) is called anIF approximation space .For any IF number (
a ,b ) ∈I ⊗I , is an IF set which has the membership value a constant ”a ” and the nonmembership value a constant ”b ” for allx ∈X .Proposition 2.4 ( [12, 13]). Let (X ,R ) be an IF approximation space. LetA ,B ∈ IF(X ), {A_{j} j ∈J } ⊆ IF(X ) and (a ,b ) ∈I ⊗I . Then we haveTheorem 2.5 ( [12, 13]). Let (X ,R ) be an IF approximation space. Then(1)
R is reflexive(2)
R is transitiveTheorem 2.6. Let (X ,R ) be a reflexive IF approximation space. If for eachj ∈J , then .Proof By the reflexivity ofR and Theorem 2.5, . By Proposition 2.4,Thus .
Example 2.7. LetX = {x _{1},x _{2}}. LetR = {<(x _{1},x _{1}), 1, 0>, <(x _{1},x _{2}), 0.2, 0.5>,<(x _{2},x _{1}), 0.4, 0.3>,<(x _{2},x _{2}), 1, 0>}. Then (X ,R ) is an IF approximation space. LetA = {<x _{1}, 0.6, 0.3>, <x _{2}, 0.5, 0.4>} be an IF set onX , thenSimilarly, we obtain
Hence,
Similarly, we have
Proposition 2.8 ( [12]). For an IF relationR onX andA ∈ IF(X ), the pair and are “dual”, i.e.,where
A^{C} is the complement ofA .Definition 2.9 ( [13, 14]). AnIF topology T onX in the sense of Lowen [15] is a family of IF sets inX that is closed under arbitrary suprema and finite infima, and contains all constant IF sets. The IF sets inT are calledopen , and their complements,closed .Definition 2.10 ( [8]). AKuratowski IF closure operator onX is a functionk : IF(X ) → IF(X ) satisfying for (a ,b ) ∈I ⊗I ,A ,B ∈ IF(X ),(i) , (ii) A ≤ k(A), (iii) k(A ∨ B) = k(A) ∨ k(B), (iv) k(k(A)) = k(A).
A Kuratowski IF closure operator
k onX is calledsaturated if for allA_{j} ∈ IF(X ),j ∈J ,Furthermore, an IF topology is called
saturated if it has a saturated IF closure operator.Remark 2.11 ( [13]). Every Kuratowski IF closure operatork onX gives rise to an IF topology onX in which an IF setB is closed iffk (B ) =B .3. Intuitionistic Fuzzy Implication Operator
Generally, for (
a _{1},a _{2}),(b _{1},b _{2}) ∈I ⊗I , theimplication operator (orresidual implicator ) [16, 17] is defined as follows;(
a _{1},a _{2}) → (b _{1},b _{2}) = sup{(d _{1},d _{2}) ∈I ⊗I  (a _{1},a _{2}) ∧ (d _{1},d _{2}) ≤ (b _{1},b _{2})}.If given IF numbers (
a _{1},a _{2}),(b _{1},b _{2}) are comparable, then theIF implication operator is clearly given byfor all (
a _{1},a _{2}),(b _{1},b _{2}) ∈I ⊗I .But it is not always able to compare given IF numbers. Nevertheless, in many papers the IF implication operator is studied where given IF numbers are comparable. In this paper, we consider the IF implication operator to the extend that the given IF numbers are not comparable with some restrictions.
Remark 3.1. The IF residual implicator is clearly given byfor all (
a _{1},a _{2}),(b _{1},b _{2}) ∈I ⊗I .Definition 3.2. (1) For (a _{1},a _{2}) ∈I ⊗I ,A ∈ IF(X ) andx ∈X , the map ((a _{1},a _{2}) ∧A ) :I ⊗I × IF(X ) → IF(X ) is defined by((
a _{1},a _{2}) ∧A )(x ) = (a _{1},a _{2}) ∧A (x ).(2) For (
a _{1},a _{2}) ∈I ⊗I ,A ∈ IF(X ) andx ∈X , the map ((a _{1},a _{2}) →A ) :I ⊗I × IF(X ) → IF(X ) is defined by((
a _{1},a _{2}) →A )(x ) = (a _{1},a _{2}) →A (x ).(3) For
A ,B ∈ IF(X ), the map (A →B ) : IF(X ) × IF(X ) → IF(X ) is defined by(
A →B )(x ) =A (x ) →B (x ), ∀x ∈X .Remark 3.3. (1) We denote by [(a _{1},a _{2}),(1, 0)] the rectangular plane which represents [a _{1}, 1]×[0,a _{2}]. For a setA ⊆X , an IF setχ_{A} :X →I ⊗I is a map defined byFrom the above, if (
a _{1},a _{2}) ∈I ⊗I andA ∈ IF(X ) are comparable, then we have that(
a _{1},a _{2}) →A =χ _{A−1[(a1, a2),(1,0)] }∨A .(2) If (
a _{1},a _{2}),(b _{1},b _{2}),(c _{1},c _{2}) ∈I ⊗I are all comparable, then(
a _{1},a _{2}) ∧ (b _{1},b _{2}) ≤ (c _{1},c _{2}) iff (b _{1},b _{2}) ≤ (a _{1},a _{2}) → (c _{1},c _{2}).Clearly the following holds;
((
a _{1},a _{2}) → (b _{1},b _{2})) ∧ ((b _{1},b _{2}) → (c _{1},c _{2})) ≤ ((a _{1},a _{2}) → (c _{1},c _{2})).Theorem 3.4. The implication operator ”→” is an IF preorder onI ⊗I .Proof By the above property and the fact(
a _{1},a _{2}) → (a _{1},a _{2}) = (1, 0),it follows.
4. Intuitionistic Fuzzy Preorder and Intuitionistic Fuzzy Topology
Let (
X , ≤) be a preordered space andA ⊆X . Let ↑A = {y ∈X y ≥x , for somex ∈A }. If ↑A =A , thenA is called anupper set . Dually ifB =↓B = {y ∈X y ≤x , for somex ∈B }, thenB is called alower set . The family of all the upper sets ofX is clearly a topology onX , which is called theAlexandrov topology (See [18]) onX , and denoted Γ(≤). We write simply Γ(X ) for the topological space (X , Γ(≤)).On the other hand, for a topological space (
X ,T ) andx ,y ∈X , letx ≤y ifx ∈U impliesy ∈U for any open setU ofX , or equivalently, . Then ≤ is a preorder onX , called thespecialization order (See [18]) onX . Denote this preorder by Ω(T ). We also write simply Ω(X ) for (X , Ω(T )).A function
f : (X , ≤_{1}) → (Y , ≤_{2}) between two preordered sets is calledorderpreserving ifx ≤_{1}y impliesf (x ) ≤_{2}f (y ).From now on we are going to enlarge the above ideas to the IF theories in a natural way.
Definition 4.1. Let (X ,R ) be an IF preordered space. ThenA ∈ IF(X ) is called anIF upper set in (X ,R ) ifA (x ) ∧R (x ,y ) ≤A (y ), ∀x ,y ∈X .Dually,
A is called anIF lower set ifA (y ) ∧R (x ,y ) ≤A (x ) for allx ,y ∈X .Let
R be an IF preorder onX . Forx ,y ∈X , the real numberR (x ,y ) can be interpreted as the degree to whichx is less than or equal toy . The conditionA (x ) ∧R (x ,y ) ≤A (y ) can be interpreted as the statement that ifx is inA andx ≤y theny is inA . Particularly, ifR is an IF equivalence relation, then an IF setA is an upper set in (X ,R ) if and only if it is a lower set in (X ,R ).The classical preorder relation
x ≤y can be naturally extended toR (x ,y ) = (1, 0) in IF preorder relation. Since (1, 0) =R (x ,y ) ≤A (x ) →A (y ),A (x ) ≤A (y ) for any IF upper setA . That is,x ≤y meansA (x ) ≤A (y ). Obviously, the notion of IF upper sets and IF lower sets agrees with that of upper sets and lower sets in classical preordered space.Definition 4.2. A functionf : (X ,R _{1}) → (Y ,R _{2}) between IF preordered spaces is calledorderpreserving ifR _{1}(x ,y ) ≤R _{2}(f (x ),f (y )), ∀x ,y ∈X .Definition 4.3. LetX be a totally ordered set. An IF setA onX is called(i) increasing if for all x, y ∈ X with x < y, A(x) ≤ A(y), (ii) decreasing if for all x, y ∈ X with x < y, A(x) ≥ A(y), (iii) monotone if A is increasing or decreasing.
Definition 4.4. LetA = (µ_{A} ,ν_{A} ) be an IF set.A is calledsimple ifµ_{A} orν_{A} is a constant function.Remark 4.5. (1) If indeterminancy degreeπ_{A} is a constant functiont , then for any two different elementsx ,y ∈X ,π_{A} (x ) =π_{A} (y ) =t . So, ifµ_{A} (x ) ≤µ_{A} (y ), thenν_{A} (x ) ≥ν_{A} (y ). HenceA (x ) ≤A (y ). ThereforeA (x ) andA (y ) are comparable for anyx ,y ∈X provided that the hesitancy degree of an IF setA is a constant function.(2) Suppose that the universal set
X is a totally ordered set. IfA is monotone or simple, thenA (x ) andA (y ) are comparable for any different elementx ,y ∈X .From now on, in order to avoid getting imprecise value in acting with the implication operator, we will consider only the IF set
A :X →I ⊗I such thatA (x ) andA (y ) are comparable for anyx ,y ∈X . By Remark 4.5, any function which is monotone or simple or of constant hesitancy degree is an example of such function.Lemma 4.6. For a given IF preordered space (X ,R ), an IF setB :X →I ⊗I is an IF upper set of (X ,R ) if and only ifB : (X ,R ) → (I ⊗I , →) is an orderpreserving function.Proof For anyB ∈ IF(X ), we have the following relations.B is an IF upper set in (X ,R )⇔ B(x) ∧ R(x, y) ≤ B(y) for all x, y ∈ X ⇔ R(x, y) ≤ B(x) → B(y) for all x, y ∈ X ⇔ The map B : (X, R) → (I ⊗ I, →) defined by preserves order.
Theorem 4.7. If (X ,R ) is an IF preordered space, then the familyT of all the upper sets inX satisfies the following conditions, and hence it is an IF topology onX .For any IF sets
A_{j} ,A ∈T ;(i) , (ii) , (iii) , (iv) , (v) Suppose that for any A ∈ T and x, y ∈ X and (a, b) ∈ I ⊗ I, A(x) and A(y) and (a, b) are all comparable. Then ((a, b) → A) ∈ T .
Proof (i) Let (a ,b ) ∈I ⊗I , then . So .(ii) .
(iii) .
(iv) If
A is an upper set, thenA (x ) ∧R (x ,y ) ≤A (y ) for anyx ,y ∈X . So, we obtain that((
a ,b ) ∧A (x )) ∧R (x ,y ) ≤ ((a ,b ) ∧A (y )),for all (
a ,b ) ∈I ⊗I . This means that ((a ,b ) ∧A ) ∈T .(v) By Remark 3.3, ((
a ,b ) →A (x )) ∧ (A (x ) →A (y )) ≤ ((a ,b ) →A (y )). Thus((a ,b ) →A (x )) → ((a ,b ) →A (y )) ≥ (A (x ) →A (y )) ≥R (x ,y ). Hence ((a ,b ) →A (x )) ∧R (x ,y ) ≤ ((a ,b ) →A (y )).Definition 4.8. For an IF preordered space (X ,R ), letIf
T is a family of IF sets inX which satisfies conditions (i)(v) of Theorem 4.7, then there exists an IF preorderR onX such thatT consists of all the upper sets of the IF preordered space (X ,R ). It will be shown in the following theorem.Lemma 4.9. Let Λ be a subfamily of IF(X ) such that for anyA ∈ Λ and for anyx ,y ∈X ,A (x ) andA (y ) are comparable. Let . ThenR is an IF preorder.Theorem 4.10. Let Λ be a subfamily of IF(X ) satisfying (i)(v) of Theorem 4.7 such that for anyA ∈ Λ and for anyx ,y ∈X ,A (x ) andA (y ) are comparable. Then there exists an IF preorderR such that Λ is the familyT of all upper sets inX with respect toR .Proof Suppose that Λ ⊆ IF(X ) satisfy the conditions (i)(v) of Theorem 4.7. Define . By the above lemma,R is an IF preorder onX . LetB ∈ Λ. SinceB (x ) andB (y ) are comparable for anyx ,y ∈X , we haveR (x ,y ) ≤B (x ) →B (y ). HenceB (x ) ∧R (x ,y ) ≤B (y ), i.e.B is an IF upper set. Thus Λ ⊆T .What remains is to show that
T ⊆ Λ. TakeD ∈T . For a givenx ∈X , definem_{x} :X →I ⊗I bym_{x} (z ) =D (x ) ∧R (x ,z ) for allz ∈X . Thenm_{x} (x ) =D (x ) andm_{x} (z ) ≤D (z ) for allz ∈X . Thus .For each
B ∈ Λ and previously givenx , defineg_{B} :X →I ⊗I byg_{B} (z ) =B (x ) →B (z ) for allz ∈X . By (v),g_{B} ∈ Λ. By (iii), we obtain that . Sincewe have
m_{x} ∈ Λ. Note that . ThereforeD ∈ Λ by (ii).Example 4.11. LetX = [0, 1] be the universal set. Let Λ be the family of all constant IF sets onX and the IF setA = (µ_{A} ,ν_{A} ), whereµ_{A} (x ) = 1−x ,ν_{A} (x ) =x . Take Γ by arbitrary suprema and arbitrary infima with members of Λ. Then clearly Γ ⊆ IF(X ) and it satisfies (i)(v) of Theorem 4.7. We can define the orderR onX byThen clearly
R is reflexive. Takex ,y ∈X such thatx ≤y , thenR (x ,y ) =A (y ) andR (y ,x ) = (1, 0). So we know thatR is transitive. ThereforeR is a preorder onX . ConsiderA (x )∧R (x ,y ) whenx ≤y . ThenR (x ,y ) =A (y ), and henceA (x ) ∧R (x ,y ) =A (x ) ∧A (y ) ≤A (y ). Consequently we know that Γ is the family of all upper sets inX with respect toR .The following result relates lower sets and upper sets in an IF preordered space (
X ,R ) with the upper approximation operator and , respectively.Proposition 4.12. Let (X ,R ) be an IF preordered space andA ∈ IF(X ). ThenA is a lower set iff .Proof A is a lower setProposition 4.13. Let (X ,R ) be an IF preordered space andA ∈ IF(X ). ThenA is an upper set iff .Proof A is an upper setRemark 4.14. LetT be the family of all upper sets of an IF preordered space (X ,R ). By Theorem 4.7,T is an Alexandrov IF topology onX . Furthermore by the above proposition.Proposition 4.15. Let (X ,R ) be an IF preordered space. Then is an Alexandrov IF topology onX .Proof (1) Take (a ,b ) ∈I ⊗I , then we know that . So . Clearly . Hence . Therefore for any (a ,b ) ∈I ⊗I .(2) Take . Then for each
j ∈J . So . Thus .(3) Take . Then for each
j ∈J . So . Thus .Remark 4.16. The IF topology is dual to the IF topology . It follows from the fact that for . In addition, the IF topologies and are Alexandrov IF topologies.Proposition 4.17. Let (X ,R ) be an IF preordered space andA ∈ IF(X ). Then(i) is the IF interior operator for the IF topology , (ii) is the IF closure operator for the IF topology .
Proof (i) We will show that for anyA ∈ IF(X ). Since , . Since , . By Theorem 2.6, . So . On the other hand, by , we obtain . Hence . Therefore .(ii) We will show that for any
A ∈ IF(X ). Since , . By the duality, . Hence .Proposition 4.18. Letk be a saturated IF closure operator onX . Then there exists an IF preorderR onX such thatk = iff(i) for any Aj ∈ IF(X), and (ii) k((a, b) ∧ A) = (a, b) ∧ k(A) for any A ∈ IF(X) and (a, b) ∈ I ⊗ I.
Proof Suppose thatk satisfies (i) and (ii). By usingk , we define an IF relationR onX asR (x ,y ) =k (χ _{{y}})(x ),x ,y ∈X .For each
A ∈ IF(X ), ifx ≠y , we have(
χ _{{y}}∧A (y ))(x ) = (χ _{{y}})(x )∧A (y ) = (0, 1)∧A (y ) = (0, 1).So , hence .
For every
x ∈X , we havewhich implies .
Conversely suppose the assumptions. Since
k is a Kuratowski IF closure operator,T = {A^{C} ∈ IF(X ) k (A ) =A } is an IF topologyT onX , and it satisfies (i). By Lemma 4.9, there exists an IF preorderR onX with respect to the familyT . Since ,k satisfies (ii) by Proposition 2.4.

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